Biological Biological Outline Contagion Contagion Biological Contagion Introduction Introduction Principles of Complex Systems Simple disease Simple disease spreading models spreading models Course 300, Fall, 2008 Background Background Prediction Introduction Prediction References References Prof. Peter Dodds Simple disease spreading models Department of Mathematics & Statistics Background University of Vermont Prediction References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/58 Frame 2/58 Biological Biological Contagion Contagion Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Background Background Naturomorphisms Prediction Prediction A confusion of contagions: References References ◮ “The feeling was contagious.” ◮ Is Harry Potter some kind of virus? ◮ “The news spread like wildfire.” ◮ What about the Da Vinci Code? ◮ “Freedom is the most contagious virus known to ◮ Does Sudoku spread like a disease? man.” ◮ Religion? —Hubert H. Humphrey, Johnson’s vice president ◮ Democracy...? ◮ “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 3/58 Frame 4/58
Biological Biological Social contagion Social contagion Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Eric Hoffer, 1902–1983 Background Background Prediction Prediction There is a grandeur in the uniformity of the mass. When a References References fashion, a dance, a song, a slogan or a joke sweeps like Optimism according to Ambrose Bierce: ( ⊞ ) wildfire from one end of the continent to the other, and a The doctrine that everything is beautiful, including what is hundred million people roar with laughter, sway their ugly, everything good, especially the bad, and everything bodies in unison, hum one song or break forth in anger right that is wrong. ... It is hereditary, but fortunately not and denunciation, there is the overpowering feeling that contagious. in this country we have come nearer the brotherhood of man than ever before. ◮ Hoffer ( ⊞ ) was an interesting fellow... Frame 5/58 Frame 6/58 Biological Biological The spread of fanaticism Imitation Contagion Contagion Introduction Introduction Hoffer’s acclaimed work: Simple disease Simple disease spreading models spreading models “The True Believer: Background Background Prediction Prediction Thoughts On The Nature Of Mass Movements” (1951) [3] References References “When people are free Quotes-aplenty: to do as they please, they usually imitate ◮ “We can be absolutely certain only about things we each other.” do not understand.” —Eric Hoffer ◮ “Mass movements can rise and spread without belief “The Passionate State in a God, but never without belief in a devil.” of Mind” [4] ◮ “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” despair.com Frame 7/58 Frame 8/58
Biological Biological The collective... Contagion Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Definitions Background Background Prediction Prediction ◮ (1) The spreading of a quality or quantity between References References individuals in a population. “Never Underestimate ◮ (2) A disease itself: the Power of Stupid the plague, a blight, the dreaded lurgi, ... People in Large ◮ from Latin: con = ‘together with’ + tangere ‘to touch.’ Groups.” ◮ Contagion has unpleasant overtones... ◮ Just Spreading might be a more neutral word ◮ But contagion is kind of exciting... despair.com Frame 9/58 Frame 10/58 Biological Biological Examples of non-disease spreading: Contagions Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Background Background Prediction Prediction References References Two main classes of contagion Interesting infections: 1. Infectious diseases: ◮ Spreading of buildings in the US. ( ⊞ ) tuberculosis, HIV, ebola, SARS, influenza, ... ◮ Spreading of spreading ( ⊞ ). 2. Social contagion: ◮ Viral get-out-the-vote video. ( ⊞ ) fashion, word usage, rumors, riots, religion, ... Frame 11/58 Frame 12/58
Biological Biological Mathematical Epidemiology Mathematical Epidemiology Contagion Contagion Introduction Introduction Simple disease Simple disease The standard SIR model [8] spreading models spreading models Discrete time automata example: Background Background ◮ The basic model of disesase contagion Prediction Prediction 1 − βI References References ◮ Three states: S 1. S = Susceptible Transition Probabilities: βI 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory β for being infected given ρ ◮ S ( t ) + I ( t ) + R ( t ) = 1 I contact with infected r for recovery ◮ Presumes random interactions (mass-action r 1 − r ρ for loss of immunity principle) R ◮ Interactions are independent (no memory) 1 − ρ ◮ Discrete and continuous time versions Frame 14/58 Frame 15/58 Biological Biological Mathematical Epidemiology Independent Interaction models Contagion Contagion Differential equations for continuous model Introduction Introduction Simple disease Simple disease spreading models spreading models d Background Background d t S = − β IS + ρ R Prediction Prediction References References Original models attributed to d d t I = β IS − rI ◮ 1920’s: Reed and Frost d ◮ 1920’s/1930’s: Kermack and McKendrick [5, 7, 6] d t R = rI − ρ R ◮ Coupled differential equations with a mass-action principle β , r , and ρ are now rates. Reproduction Number R 0 : ◮ R 0 = expected number of infected individuals resulting from a single initial infective ◮ Epidemic threshold: If R 0 > 1, ‘epidemic’ occurs. Frame 16/58 Frame 17/58
Biological Biological Reproduction Number R 0 Reproduction Number R 0 Contagion Contagion Discrete version: Introduction Introduction Simple disease Simple disease ◮ Expected number infected by original Infective: Discrete version: spreading models spreading models Background Background Prediction Prediction ◮ Set up: One Infective in a randomly mixing R 0 = β + ( 1 − r ) β + ( 1 − r ) 2 β + ( 1 − r ) 3 β + . . . References References population of Susceptibles ◮ At time t = 0, single infective random bumps into a � 1 + ( 1 − r ) + ( 1 − r ) 2 + ( 1 − r ) 3 + . . . � = β Susceptible ◮ Probability of transmission = β 1 ◮ At time t = 1, single Infective remains infected with = β 1 − ( 1 − r ) = β/ r probability 1 − r ◮ At time t = k , single Infective remains infected with For S 0 initial infectives (1 − S 0 = R 0 immune): probability ( 1 − r ) k R 0 = S 0 β/ r Frame 18/58 Frame 19/58 Biological Biological Independent Interaction models Independent Interaction models Contagion Contagion For the continuous version Introduction Introduction Example of epidemic threshold: Simple disease Simple disease ◮ Second equation: 1 spreading models spreading models Background Background Prediction Prediction 0.8 Fraction infected d References References d t I = β SI − rI 0.6 0.4 d d t I = ( β S − r ) I 0.2 0 0 1 2 3 4 ◮ Number of infectives grows initially if R 0 β S ( 0 ) − r > 0 ⇒ β S ( 0 ) > r ⇒ β S ( 0 ) / r > 1 ◮ Continuous phase transition. ◮ Fine idea from a simple model. ◮ Same story as for discrete model. Frame 20/58 Frame 21/58
Biological Biological Independent Interaction models Disease spreading models Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Background Background Prediction Prediction Many variants of the SIR model: References References ◮ SIS: susceptible-infective-susceptible For novel diseases: ◮ SIRS: susceptible-infective-recovered-susceptible 1. Can we predict the size of an epidemic? ◮ compartment models (age or gender partitions) 2. How important is the reproduction number R 0 ? ◮ more categories such as ‘exposed’ (SEIRS) ◮ recruitment (migration, birth) Frame 22/58 Frame 24/58 Biological Biological R 0 and variation in epidemic sizes Size distributions Contagion Contagion Introduction Introduction Simple disease Simple disease spreading models spreading models Background Background Size distributions are important elsewhere: Prediction Prediction References References R 0 approximately same for all of the following: ◮ earthquakes (Gutenberg-Richter law) ◮ 1918-19 “Spanish Flu” ∼ 500,000 deaths in US ◮ city sizes, forest fires, war fatalities ◮ 1957-58 “Asian Flu” ∼ 70,000 deaths in US ◮ wealth distributions ◮ 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US ◮ ‘popularity’ (books, music, websites, ideas) ◮ 2003 “SARS Epidemic” ∼ 800 deaths world-wide ◮ Epidemics? Power laws distributions are common but not obligatory... Frame 25/58 Frame 26/58
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